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Liquidxlax
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Homework Statement
consider a sigle loop of the cycloid with a fixed value of a. A car is released at a point P0 from rest anywhere on the track between the origin and the lowest point P, that is P0 has a parameter 0<theta0 < pi. show that the time for the cart to roll from P0 to P is given by the integral
time( P0]/sub] -> P) = [itex]\sqrt{\frac{a}{g}}\int \sqrt{\frac{1 - cos\vartheta}{cos\vartheta_{0} - cos\vartheta}}d\vartheta[/itex]
integral is from theta naught to pi
and prove the integral equals [itex]\pi\sqrt{\frac{a}{g}}[/itex] the integral may be tricky and you can use theta = pi -2(alpha)
Homework Equations
[itex]\frac{df}{dx}=\frac{d}{dy}\frac{df/dx'}[/itex]
The Attempt at a Solution
1/2 mv2 = mg(y-y1
v = [itex]\sqrt{2g(y-y1}[/itex]
dt = ds/v
T = [itex]\int\frac{\sqrt{1+(x')2}}{\sqrt{2g(y-y1}}[/itex] dy
test the Euler formula to get
y = a(1 - cos(theta)) y' = asin(theta)
x = a (theta - sin(theta)) x' = a - acos(theta)
This is now the part I am having a problem with
Now i substitute my y and x' in
T = [itex]\int\frac{\sqrt{1 + (a( 1 - cos(\vartheta))2}}{\sqrt{2ga(cos(\vartheta) - cos(\vartheta0)}}(asin(\vartheta)[/itex]
and somehow that equals the top equation. Then i need to integrate it, which i get no where near close to what it wants no matter what the substitution and trig identities
not sure why all the itex isn't working